Maxwell equations plane-wave solutions

Let us look for plane-wave solutions to the Maxwell equations (2.12)- (2.15). What does this statement mean We know that the electromagnetic field (E, H) cannot be arbitrarily specified. Only certain electromagnetic fields, those that satisfy the Maxwell equations, are physically realizable. Therefore, because of their simple form, we should like to know under what conditions plane electromagnetic waves... 

Consider a plane wave propagating in a nonabsorbing medium with refractive index N2 = n2, which is incident on a medium with refractive index A, = w, + iky (Fig. 2.4). The amplitude of the incident electric field is E(, and we assume that there are transmitted and reflected waves with amplitudes E, and Er, respectively. Therefore, plane-wave solutions to the Maxwell equations at... 

Consider a non-plane-wave solution of Maxwell s equations, whose direction of propagation varies with respect to the z axis. In general it holds that... 

Up to now, we have examined how the Beltrami vector field relation surfaces in many electromagnetic contexts, featuring predominantly plane-wave solutions (PWSs) to the free-space Maxwell equations in conjunction with biisotropic media (Lakhtakia-Bohren), in homogeneous isotropic vacua (Hillion/Quinnez), or in the magnetostatic context exemplified by FFMFs associated with plasmas (Bostick, etc.). 

In the plane-wave solution of Maxwell s equations, the electric vector of light is transverse and composed of two polarization components, Ev and Eh, that are orthogonal to each other. Both the incident light and the scattered light can be fully characterized (in terms of intensity, polarization and phase) by a set of Stokes parameters, S(Iv,lh,U,V), defined by ... 

Our fundamental task is to construct solutions to the Maxwell equations (3.1)—(3.4), both inside and outside the particle, which satisfy (3.7) at the boundary between particle and surrounding medium. If the incident electromagnetic field is arbitrary, subject to the restriction that it can be Fourier analyzed into a superposition of plane monochromatic waves (Section 2.4), the solution to the problem of interaction of such a field with a particle can be obtained in principle by superposing fundamental solutions. That this is possible is a consequence of the linearity of the Maxwell equations and the boundary conditions. That is, if Ea and Efc are solutions to the field equations,... 

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

A complete set of standard time-harmonic solutions to Maxwell s equations usually involve the plane wave decomposition of the field into transverse electric... 

The solutions to the Maxwell field equations that are most often used in the applications discussed in this book are referred to as plane waves of monochromatic light. These are derived from the Maxwell curl equations. In a system free of charges and currents, these are... 

In an anisotropic dielectric the phase velocity of an electromagnetic wave generally depends on both its polarization and its direction of propagation. The solutions to Maxwell s electromagnetic wave equations for a plane wave show that it is the vectors D and H which are perpendicular to the wave propagation direction and that, in general, the direction of energy flow does not coincide with this. 

When a plane electromagnetic wave is incident on a spherical distribution of charge of radius R and dielectric constant e (which is in general complex), vibrations are set up, and an absorption cross section can be calculated from Maxwell s equations. The Mie solution for the cross section is... 

Mie Scattering. For systems more complex than very small particles (Rayleigh) or small particles with low refractive indices (Rayleigh-Debye), the scattering from widely separated spherical particles requires solving Maxwells equations. The solution of these boundary-value problems for a plane wave incident upon a particle of arbitrary size, shape, orientation, and index of refraction has not been achieved mathematically, except for spheres via the Mie theory (12,13). Mie obtained a series expression in terms of spherical harmonics for the intensity of scattered light emergent from a sphere of arbitrary size and index of fraction. The coeflBcients of this series are functions of the relative refractive index m and the dimensionless size parameter a = ird/k. 

Thus Eqs. (6.4) and (6.5) deseribe a plane electromagnetie wave propagating in a homogeneous medium without sources. This is a very important solution of the Maxwell equations beeause it embodies the eoneept of a perfectly monochromatic parallel beam of light of infinite lateral extent and represents the transport of eleetromagnetie energy from one point to another. 

In electromagnetic theory the Faraday effect can be explained as follows. When the medium magnetization has non-zero projection on the wave vector ko of the incident radiation, two independent fundamental Maxwell equations solutions are circular polarized waves with different refractive indexes n+M n, respectively. At the output of the magnetic medium these waves gain phase shift and when added give Unearly polarized wave with rotated polarization plane. That is why Faraday effect is also called magnetic circular birefringence [26, 27]. 

Fresnel equations relate the electric field strength amphtudes of the incident, reflected, and transmitted waves. They are solutions of Maxwells equations by applying the above-mentioned boundary conditions. It can be shown that for a plane boundary between two non-magnetic isotropic phases of infinite thickness, schematically depicted in Fig. 9.1, the Fresnel reflection (r) and transmission (t) coefficients for s- and p-polarized light are given by the following equations ... 

The full details to the solution of Maxwell s equations are outlined in several other resources [1], Here, only the results relevant to evanescent wave microscopy are presented, specifically the functionality of the intensity field in the low density material, which is used for imaging. It should be noted that these solutions assume infinite plane waves incident on the interface. In practice, however, one typically uses a finite, Gaussian laser beam, which is well approximated using these assumptions. The intensity field has the exponential form... 

For our purposes, we will not need to use Maxwell s equations themselves we can focus on a solution to these equations for a particular situation such as the plane wave of monochromatic, polarized light illustrated in Fig. 3.3. In a uniform, homogeneous, nonconducting medium with no free charges. Maxwell s equations for in a one-dimensional plane wave reduce to... 

As discussed in Box 3.1, convenient solutions to Maxwell s equations also can be obtained in terms of a vector potential V instead of electric and magnetic fields. Using the same formalism as Eq. (3.17) but omitting the phase shift for simplicity, the vector potential for a plane wave of monochromatic, linearly polarized light can be written... 

Diffractive ARS are usually calculated either using the effective medium approximation, or the rigorous coupled-wave analysis [169]. Direct numerical solution of Maxwell equations and the approach using the plane wave theory were also used [174]. 

Maxwell s equations yield a wave equation that predicts the propagation of electromagnetic radiation. The most elementary solution is the plane wave expressible as a function of space (x,y, z) and time (t) as... 

In the special case of free space, i.e. an unbounded, uniform medium of refractive-index n, any solution of Maxwell s equations is expressible as a continuous superposition of plane waves, or rays, propagating in all directions. Thus, for example, if the electric field is everywhere parallel to the x-direction on the infinite surface z = 0 and has the spatial distribution E (x, y), then [2]... 

It is useful to consider the solution of Maxwell s Equations (5.1) for plane electromagnetic waves in the absence of boundary conditions, which can be written as exp[i(/ 2 — u>t) assuming propagation in z-direction of cartesian coordinates. The quantity / is the complex propagation constant of the medium with dominant real part for dielectrics and dominant imaginary part for metals. The impedance of the medium, Z, defined as ratio of electric to magnetic field is related to / by Z = ojp,0/f3 with /x0 = 1.256 x 10 6 Vs/Am. As it can be derived from Maxwell s equations, the impedance is related to the conductivity/dielectric function by the following expression ... 

Exercise 7.1 Show that the plane-polarized electromagnetic wave in Eq. (7.1) is a solution to Maxwell s equation for the vector potential in vacuum Eq. (2.129). 

The refractive indices of anisotropic materials are conveniently represented in terms of the optical indicatrix, the surface of which maps the refractive indices of propagating waves as a function of angle. Solution of Maxwell s equations for an anisotropic medium leads to the result that for a particular wave-normal, two waves may propagate with orthogonal plane-polarisations and different refractive indices. An ellipsoid having as semi-axes the three principal refractive indices defines the optical indicatrix. In general for any wave-normal, the section of the indicatrix perpendicular to the wave-normal direction will be an ellipse, and the semi-axes of this ellipse are the refractive indices of the two propagating waves. 

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